Integrand size = 40, antiderivative size = 74 \[ \int (a+a \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=a^2 (B+2 C) x+\frac {a^2 (2 B+C) \text {arctanh}(\sin (c+d x))}{d}-\frac {a^2 (B-C) \sin (c+d x)}{d}+\frac {B \left (a^2+a^2 \cos (c+d x)\right ) \tan (c+d x)}{d} \]
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Time = 0.33 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3108, 3054, 3047, 3102, 2814, 3855} \[ \int (a+a \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\frac {a^2 (2 B+C) \text {arctanh}(\sin (c+d x))}{d}-\frac {a^2 (B-C) \sin (c+d x)}{d}+\frac {B \tan (c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{d}+a^2 x (B+2 C) \]
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Rule 2814
Rule 3047
Rule 3054
Rule 3102
Rule 3108
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \int (a+a \cos (c+d x))^2 (B+C \cos (c+d x)) \sec ^2(c+d x) \, dx \\ & = \frac {B \left (a^2+a^2 \cos (c+d x)\right ) \tan (c+d x)}{d}+\int (a+a \cos (c+d x)) (a (2 B+C)-a (B-C) \cos (c+d x)) \sec (c+d x) \, dx \\ & = \frac {B \left (a^2+a^2 \cos (c+d x)\right ) \tan (c+d x)}{d}+\int \left (a^2 (2 B+C)+\left (-a^2 (B-C)+a^2 (2 B+C)\right ) \cos (c+d x)-a^2 (B-C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx \\ & = -\frac {a^2 (B-C) \sin (c+d x)}{d}+\frac {B \left (a^2+a^2 \cos (c+d x)\right ) \tan (c+d x)}{d}+\int \left (a^2 (2 B+C)+a^2 (B+2 C) \cos (c+d x)\right ) \sec (c+d x) \, dx \\ & = a^2 (B+2 C) x-\frac {a^2 (B-C) \sin (c+d x)}{d}+\frac {B \left (a^2+a^2 \cos (c+d x)\right ) \tan (c+d x)}{d}+\left (a^2 (2 B+C)\right ) \int \sec (c+d x) \, dx \\ & = a^2 (B+2 C) x+\frac {a^2 (2 B+C) \text {arctanh}(\sin (c+d x))}{d}-\frac {a^2 (B-C) \sin (c+d x)}{d}+\frac {B \left (a^2+a^2 \cos (c+d x)\right ) \tan (c+d x)}{d} \\ \end{align*}
Time = 2.09 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.93 \[ \int (a+a \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\frac {a^2 \left (B c+2 c C+B d x+2 C d x-2 B \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+2 B \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+C \sin (c+d x)+B \tan (c+d x)\right )}{d} \]
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Time = 4.97 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.09
method | result | size |
parts | \(\frac {\left (B \,a^{2}+2 a^{2} C \right ) \left (d x +c \right )}{d}+\frac {\left (2 B \,a^{2}+a^{2} C \right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {a^{2} B \tan \left (d x +c \right )}{d}+\frac {\sin \left (d x +c \right ) a^{2} C}{d}\) | \(81\) |
derivativedivides | \(\frac {a^{2} C \sin \left (d x +c \right )+B \,a^{2} \left (d x +c \right )+2 a^{2} C \left (d x +c \right )+2 B \,a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{2} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,a^{2} \tan \left (d x +c \right )}{d}\) | \(88\) |
default | \(\frac {a^{2} C \sin \left (d x +c \right )+B \,a^{2} \left (d x +c \right )+2 a^{2} C \left (d x +c \right )+2 B \,a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{2} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,a^{2} \tan \left (d x +c \right )}{d}\) | \(88\) |
parallelrisch | \(-\frac {2 a^{2} \left (\cos \left (d x +c \right ) \left (B +\frac {C}{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\cos \left (d x +c \right ) \left (B +\frac {C}{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {\sin \left (2 d x +2 c \right ) C}{4}-\frac {d x \left (B +2 C \right ) \cos \left (d x +c \right )}{2}-\frac {B \sin \left (d x +c \right )}{2}\right )}{d \cos \left (d x +c \right )}\) | \(103\) |
risch | \(a^{2} B x +2 a^{2} C x -\frac {i {\mathrm e}^{i \left (d x +c \right )} a^{2} C}{2 d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} a^{2} C}{2 d}+\frac {2 i B \,a^{2}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {2 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{d}-\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{d}+\frac {2 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{d}+\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{d}\) | \(163\) |
norman | \(\frac {\left (B \,a^{2}+2 a^{2} C \right ) x +\left (-4 B \,a^{2}-8 a^{2} C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-B \,a^{2}-2 a^{2} C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-B \,a^{2}-2 a^{2} C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (B \,a^{2}+2 a^{2} C \right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (2 B \,a^{2}+4 a^{2} C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (2 B \,a^{2}+4 a^{2} C \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {4 a^{2} \left (B -C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 a^{2} \left (B -C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 a^{2} \left (B +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {4 a^{2} \left (B +C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 a^{2} \left (3 B -C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 a^{2} \left (3 B +C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {a^{2} \left (2 B +C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {a^{2} \left (2 B +C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(392\) |
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Time = 0.28 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.46 \[ \int (a+a \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\frac {2 \, {\left (B + 2 \, C\right )} a^{2} d x \cos \left (d x + c\right ) + {\left (2 \, B + C\right )} a^{2} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (2 \, B + C\right )} a^{2} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (C a^{2} \cos \left (d x + c\right ) + B a^{2}\right )} \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \]
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\[ \int (a+a \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=a^{2} \left (\int B \cos {\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int 2 B \cos ^{2}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int B \cos ^{3}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int C \cos ^{2}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int 2 C \cos ^{3}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int C \cos ^{4}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx\right ) \]
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Time = 0.21 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.42 \[ \int (a+a \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\frac {2 \, {\left (d x + c\right )} B a^{2} + 4 \, {\left (d x + c\right )} C a^{2} + 2 \, B a^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + C a^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, C a^{2} \sin \left (d x + c\right ) + 2 \, B a^{2} \tan \left (d x + c\right )}{2 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (74) = 148\).
Time = 0.32 (sec) , antiderivative size = 155, normalized size of antiderivative = 2.09 \[ \int (a+a \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\frac {{\left (B a^{2} + 2 \, C a^{2}\right )} {\left (d x + c\right )} + {\left (2 \, B a^{2} + C a^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - {\left (2 \, B a^{2} + C a^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1}}{d} \]
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Time = 1.30 (sec) , antiderivative size = 161, normalized size of antiderivative = 2.18 \[ \int (a+a \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\frac {C\,a^2\,\sin \left (c+d\,x\right )}{d}+\frac {2\,B\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {4\,B\,a^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {4\,C\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,C\,a^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {B\,a^2\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )} \]
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